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One Universe: At Home in the Cosmos


energy: TOC for Knowledge Concepts, Exercises, and Solutions

Two of the most fundamental questions we ask when studying any object
in the universe are: (1) how bright is it?  and (2) how far away is it?
It turns out these two quantities are connected by an INVERSE SQUARE 

The word "brightness" is unclear; does it mean how much luminous energy
an object gives off?  Or does it mean how much light you see coming from
that object?  Astronomers use two words to break the ambiguity: luminosity
and flux.

LUMINOSITY is the amount of energy an object gives off per unit time.
This quantity is often given in units of joules/second, otherwise known
as Watts (W).  This is the same unit that's used in electric circuits,
but in the form of radiated energy rather than electric power.

   e.g. The Sun's luminosity is 3.9 * 10^26 W
   e.g. A 60-watt light bulb requires 60 W of electric power to
        operate, but only gives off about 6 W of luminosity in visible
        light!  (Rather inefficient...)

FLUX is the amount of energy from a luminous object that reaches a given
surface or location.  This quantity is often given in watts per square
meter (W/m^2).  This is how bright an object appears to the observer.

   e.g. The Sun's flux on Earth is about 1400 W/m^2 

Luminosity and flux are related mathematically.  We can visualize this
relationship: imagine a light source that has a luminosity of 100 W,
enveloped by a round balloon that has a surface area of one m^2.  Then
that balloon receives a total flux of 100 W/m^2.  If the light source is
shining equally brightly in every direction, then every spot on that
balloon receives a flux of 100 W/m^2.  In other words, the flux is
given by the luminosity divided by the surface area of the balloon.

A sphere's surface area is related to its radius by A = (4 * pi * r^2).
So imagine a sphere of radius r surrounding an object with luminosity L.
Any object on the surface of that sphere, then, would receive a flux

      F = ------------  
          4 * pi * r^2

(Remember that pi = 3.1416 or so.)  Notice the r^2 in the denominator;
so if you double r, the flux is 1/r^2 or one-fourth what it was.
   If you triple r, the flux is 1/r^2 or one-ninth what it was.
     " quadruple r, the flux is 1/r^2 or one-sixteenth what it was.

This kind of relationship is called an INVERSE SQUARE law.  To compute
the flux from any object onto any observer, visualize the distance 
between the object and the observer as the radius of a large sphere.

   e.g. If you're standing 2 meters away from a light source
        that has a luminosity of 9 watts, the flux on you is

               9 W
     F = ------------------ = 0.18 W/m^2
         4 * 3.14 * (2 m)^2	 4 * 3.14 * (2 m)^2

Astronomers usually measure the flux of distant objects onto their
telescopes - this is the basic thing we do with telescopes, in fact!
So if we also know the object's distance, we can compute its luminosity,
and vice versa.